Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Beat butter, 1/2 cup white sugar, brown sugar, and egg in a large bowl until smooth. Stir in flour, oats, 1/3 cup peanut butter, baking soda, vanilla extract, and salt until dough is well mixed. Press dough into prepared baking dish.
Bake in preheated oven until golden brown, about 20 minutes; sprinkle with chocolate chips immediately.
Let chocolate chips melt, about 5 minutes.
Spread chocolate over cookie with a spatula or knife.
Mix confectioners' sugar, 1/4 cup peanut butter, and milk in a small bowl until smooth.
Add more milk if mixture is too thick to pour; drizzle over chocolate and allow to cool.